The present invention relates to a mathematical analysis for the quantitative estimation of the level of differential gene expression. More specifically, the present invention relates to the mathematical derivation of an a posteriori distribution of all the fold-changes of the level of gene expression which may be inferred from the given experimental measurements.
Cells rely upon their numerous protein components for a wide variety of functions. These functions include, e.g., the production of energy, the biosynthesis of all component macromolecules, the maintenance of cellular architecture, the ability to act upon intra- and extracellular stimuli, and the like. Each cell within an organism in contains the information necessary to produce the repertoire of proteins that organism can expressed. This information is stored as genes within the organism""s genome. The number of unique human genes is estimated to be 30,000 to 100,000.
For any given cell, only a portion of the gene set is expressed in the form of protein. Some of the proteins are likely to be present in all cells (i.e., are ubiquitously expressed) because they serve biological function(s) which are required in every type of cell, and can be thought of as xe2x80x9chousekeepingxe2x80x9d proteins. In contrast, other proteins serve specialized functions that are only required in particular cell types. For example, muscle cells contain specialized proteins that form the dense contractile fibers of a muscle. Given that a large part of a cell""s specific functionality is determined by the genes it is expressing, it is logical that transcription, the first step in the process of converting the genetic information stored in an organism""s genome into protein, would be highly regulated by the control network that coordinates and directs cellular activity.
Regulation of gene expression is readily observed in studies that examine activities evident in cells configuring themselves for a particular function (e.g., specialization into a muscle cell) or state (e.g., active multiplication or quiescence). Hence, as cells alter their status, the coordinated transcription of the protein(s) which are requited for this molecular biological/physiological xe2x80x9cstatexe2x80x9d can be observed. This highly detailed, global knowledge of the cell""s transcriptional state provides information on the cell""s status, as well as on the biological system(s) controlling this status. For example, knowledge of when and in what types of cell the protein product of a gene of unknown function is expressed would provide useful clues as to the likely function of that gene. Determination of gene expression patterns in normal cells could provide detailed knowledge of the way in which the control system achieves the highly coordinated activation and deactivation required for development and differentiation of a mature organism from a single fertilized egg. Comparison of gene expression patterns in normal and pathological cells could provide useful diagnostic xe2x80x9cfingerprintsxe2x80x9d and help identify aberrant functions that would be reasonable targets for therapeutic intervention.
Unfortunately, the ability to carry out studies in which the transcriptional state of a large number of genes is determined has, until recently, been inhibited by limitations on the ability to survey cells for the presence and abundance of a large number of gene transcripts in a single experiment. One limitation can be in the small number of identified genes. In the case of humans, only a few thousand proteins encoded within the human genome have been physically purified and quantitatively characterized to any extent. Another limitation can be in the manner of transcription analysis.
Two recent technological advances address have aided analyses of gene transcription. The cloning of molecules derived from mRNA transcripts in particular tissues, followed by the application of high-throughput sequencing to the DNA ends of the members of these libraries has yielded a catalog of expressed sequence tags (ESTs). See e.g., Boguski and Schuler, Nat. Genetics 10: 369-370 (1995). These xe2x80x9csignature sequencesxe2x80x9d can provide unambiguous identifiers for a large cohort of genes.
In addition, the clones from which these sequences were derived provide analytical reagents that can be used in the quantitation of transcripts front biological samples. The nucleic acid polymers, DNA and RNA, are biologically synthesized in a copying reaction in which one polymer serves as a template for the synthesis of an opposing strand, which is termed its complement. Following the separation of the strands from one another (i.e., denaturation), these strands can be induced to pair, quite specifically, with other nucleic acid strands possessing a complementary sequence in a process called hybridization. This specific binding can be the basis of analytical procedures for measuring the amount of a particular species of nucleic acid, such as the mRNA specifying a particular protein gene product.
A second advance involves microarray/microassay technology. This is a hybridization-based process which allows simultaneous quantitation of many nucleic acid species. See e.g., DeRisi et al., Nat. Genetics 14: 457-460 (1996); Schena et al., Proc. Natl. Acad. Sci. USA 93: 10614-10619 (1996). This technique combines robotic placement (i.e., spotting) of small amounts of individual, pure nucleic acid species on a glass surface, hybridization to this array with multiple fluorescently-labeled nucleic acids, and detection and quantitation of the resulting fluorescent-labeled hybrids with, for example, a scanning confocal microscope. When used to detect transcripts, a particular RNA transcript (i.e., an mRNA) can be copied into DNA (i e., a cDNA) and this copied form of the transcript is subsequently immobilized onto, for example, a glass surface.
A problem in the analysis of gene expression data is the estimation of the overall fold-change in the expression level of a gene in one experiment relative to its expression in another experiment. Given these two raw measurements of the fold-change in gene expression level, the simplest approach, as utilized by previous methodologies, has been to take the arithmetic ratio of the values as an estimate of the overall fold-change. While for very strong signals this leads to a meaningful estimate of the fold-change in the underlying mRNA concentrations, for weaker signals the results are much more ambiguous because of contamination by the xe2x80x9cnoisexe2x80x9d which is indigenous to the particular experimental system utilized. Another previously utilized technology for the estimation of the fold-change in gene expression level is based upon differential-signal intensities (e.g., the Affymetrix(copyright) chip). However, the values assigned to expression levels by use of the aforementioned methodology can be negative, thus leading to the awkward situation of negative or undefined gene expression ratios.
The present invention provides a highly accurate and reproducible mathematically-based methodology for quantifying the levels of differential gene expression from microassay protocols.
The methods of the present invention can be used to calculate differences in the level of gene expression in one or more arrays of genes. The methods involve defining the experimental noise associated with intensity of hybridization signal for each gene in the array(s). The experimental noise is variations in observed levels on chips or other microarrays rather than biological noise, which is the variation of expression level seen in biological systems. Detection of genes is often, but not always, based on fluorescence. Other detection systems have been used which may be adapted here. Such systems include luminescent or radioactive labels, biotinylated, haptenated, or other chemical tags that allow for easy detection of labeled probes.
For a mathematical description, see Section Ixe2x80x94Formulation of the Noise Model below. The noise is assumed to be Gaussian and Bayes Theorem is applied. The defined experimental noise term, sigma, is then used to define an analytical (i.e., analytical in the mathematical sense meaning that it is a continuous function) probability distribution function (xe2x80x9cpdfxe2x80x9d) describing distribution values of intensity for each gene. These pdfs are used to derive an analytical joint pdf describing possible ratios or fold changes for any differentially-expressed gene or gene product in the array(s). The joint pdfs are applied using experimentally-derived intensities and noise values from the genes on the array(s) (1) to estimate fold changes in the concentration of gene transcripts, (2) to use the jpdf to establish the confidence limits on the fold change given specific confidence intervals, and (3) to derive a p-value, or quality metric (the probability a fold change could be less than 1, when the estimate is greater than 1, or, the probability that the fold change is greater than 1, when the estimate is less than 1), associated with the fold change estimate. The estimated fold change determined by the methods of the present invention represents the difference in level of gene expression observed. The total variance (i.e., noise) may still be high even as the concentration of transcript goes to zero. The methods of the present invention use a mathematical formula to describe an a posteriori statistical distribution of all the levels of gene expression which may be derived from the measurements obtained of levels of gene expression in one or more cells or tissue types represented in the array(s).
Microarrays are an ordered array of double stranded or single stranded DNA molecules positioned on a support material in a spatially separated organization. In contrast to filter xe2x80x9cmacroarraysxe2x80x9d, which are typically large sheets of nitrocellulose, microarrays position the DNA more densely packed organization such that up to 10000 DNA molecules can be fit into a region typically 1-4 square centimeters. Microarrays typically use coated glass as the solid support, in contrast to the nitrocellulose-based material of filter arrays. By having an ordered array of DNA samples, the position of each sample can be tracked and linked to the original sample from which the DNA on the array was generated. Methods and apparatus for preparing a microarray have been described. See, e.g., U.S. Pat. Nos. 5,445,934 and 5,800,992, both incorporated herein by reference.
The DNA samples on the microarray are hybridized with RNA or DNA probes that have been fluorescently labeled to identify whether the probe sample contains a molecule that is similar or identical to the DNA sample on the microarray. Under the appropriate conditions, probe molecules hybridize to a DNA molecule on the microarray. Generally, identical or near identical sequences form productive hybrids. The presence of DNA-probe hybrid molecules is detected by a fluorescence detection instrument. If the hybridization signal is weak or non-existent at a particular DNA site, then the corresponding DNA or RNA molecule in the probe is absent. Current microarray instruments can hybridize up to four different fluorescent probe samples at one time. With improvements to the technology, more probes can be hybridized at once.
Up until recently, DNA hybridizations were performed on nitrocellulose filters. In contrast to microarrays where DNA is spotted directly onto the microarray, filter arrays are generated by spotting bacterial colonies on the filters, placing the filters over a agar growth media, and incubating the filters under conditions that promote the bacterial colonies to grow. The DNA within the bacterial colonies is released by lysing the colony and treating the filters to fix the DNA to the filter material. The process of generating a bacterial filter array can take typically 2-4 days. Microarrays have a number of advantages over filter array methods. For example, filter methods generally array bacterial colonies in which the cloned cDNA is contained. The colonies must be grown up over several days, lysed to release DNA and fix DNA onto the filter. Hybridization to filter arrays of colonies is less reliable due to bacterial debris and the low amount of DNA released from the colony. A second advantage is that the iterations are quicker with microarrays than with filters. This is due to the time needed to grow colonies on the filters and prepare them for the next round of hybridization. In contrast, probing of a subsequent microarray can begin less than 24 hr after analysis of an array is completed. Another advantage of microarrays is the ability to use fluorescently labeled probes. This provides for a non-radioactive method for hybridization detection. In contrast, filter hybridization generally uses probes labeled with radioactive phosphorus or sulfur. Microarrays can be hybridized with multiple probes simultaneously. In contrast, filter arrays can only be hybridized with one probe at a time. One of the most important advantages of microarrays is their reproducibility and sensitivity of hybridization signals. Typically, hybridization signals are higher and sensitivity is greater on microarrays versus filter arrays. In addition, filter arrays often exhibit spurious background signals that are unrelated to productive hybridization between the probe and DNA on the filter.
Once the random sample of nucleic acid fragments is immobilized to a solid surface (e.g., glass) in a microarray, the random sample of nucleic acid fragments can then be hybridized to one or more labeled probes complementary to genes or sequences of interest. Generally, the unhybridized probes are removed. The labeled probes are then detected by methods known in the art (e.g., confocal microscopy). For example, slide images can be analyzed with Array Vision image analysis software (Imaging Research) for spot finding analysis, localized background determination, distribution of signal intensities in a spot, and signal to noise ratios. Statistical assessment is then performed as described below.
The present invention utilizes a mathematically-based methodology to quantitative the fold-change in the levels of differentially expressed genes. Specifically, the present invention uses a simple deductive approach, grounded in a Bayesian framework, to circumvent the heuristic-based limitation of previous methodologies used in the mathematical analysis of differential gene expression. The present invention, rather than immediately seeking a point-estimate of the fold-change of the level of gene expression, derives a mathematical formula for the a posteriori distribution of all the fold-changes of differential gene expression which may be inferred from the given measurements. From this a posteriori distribution the following information may be obtained: (i) an estimator for the fold-change of the level of gene expression; (ii) confidence limits for the fold-change, at any given confidence level; and (iii) a P-value for assessing the statistical significance of change. An additional advantage of the present invention is that fold-change estimates and confidence limits may even be assigned to signal pairs where both signals are zero or negative, without resorting to heuristic thresholds. Hence, the mathematical framework disclosed herein unifies estimation for all signals within a given sample.